# ch04_06 - 4-51 SECTION 4.6 2. When solving differential...

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4-51SECTION 4.6..Integration by Substitution3932.When solving differential equations of the formdydt=f(y)for the unknown functiony(t), it is often convenient to makeuse of apotential functionV(y). This is a function such thatdVdy=f(y). For the functionf(y)=yy3, find a potentialfunctionV(y). Find the locations of the local minima ofV(y)and use a graph ofV(y) to explain why this is called a “double-well” potential. Explain each step in the calculationSincedVdt0, does the functionVincrease or decrease astime goes on? Use your graph ofVto predict the possible val-ues of limt→∞y(t). Thus, you can predict the limiting value ofthe solution of the differential equation without ever solvingthe equation itself. Use this technique to predict limt→∞y(t) ify=22y.
EXAMPLE 6.1Finding an Antiderivative by Trial and ErrorEvaluate2xex2dx.
394CHAPTER 4..Integration4-52Note that, in general, ifFis any antiderivative off, then from the chain rule, wehaveddx[F(u)]=F(u)dudx=f(u)dudx.From this, we have thatf(u)dudxdx=ddx[F(u)]dx=F(u)+c=f(u)du,(6.1)sinceFis an antiderivative off. If you read the expressions on the far left and the far rightsides of (6.1), this suggests thatdu=dudxdx.

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